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4.4 Parameter Uncertainty
4.4.1 Parameter Uncertainty Propagation
Typically, fire models are run using a discrete set of input parameters that describe a single, specific fire scenario. However, for some Fire PRA applications, it may be necessary to consider the range of consequences due to the variability that can result from that specific fire scenario within a particular compartment. If the key input parameters can be expressed in the form statistical distributions, then the model output quantities may also be expressed as
UNCERTAINTY AND SENSITIVITY distributions. In this way, it is possible to determine the probability of exceeding a critical
temperature, heat flux, or some other critical value.
Notarianni and Parry (SFPE Handbook, 4th edition) discuss a number of techniques for
propagating parameter uncertainty. The most common are Monte Carlo methods in which the fire model is run repeatedly with randomly chosen input parameters, based on specified probability distributions. For simple algebraic models, this technique is relatively simple and there are various software packages available to help. For zone models, the technique
becomes more complicated because it requires more time to set up and run the model, but it is still practical if the number of parameters is reduced and the ranges of the parameters are appropriately discretized into “bins”. For CFD models, the technique is not practical except in special cases where the number of model runs can be reduced to a relatively small number.
The increased accuracy afforded by the CFD model is often unwarranted given that the uncertainty in the input parameters is typically greater than that of the models themselves.
Because the HRR is the most important input parameter in most fire model analyses, and because NUREG/CR-6850 (EPRI 1011989) provides distributions of the HRR for a variety of combustibles within an NPP, parameter uncertainty propagation for fire modeling may involve only the HRR distribution applied within an algebraic model. In fact, Appendix E of NUREG/CR-6850 (EPRI 1011989) provides data which can be used to illustrate a simple technique to propagate the HRR distribution.
Suppose, for example, that as part of a Fire PRA the problem is to determine the probability of flames extending above an electrical cabinet to a particular height, threatening a cable tray. To answer this question, the flame height needs to be represented as a probability distribution.
Figure 4-4 displays the distribution13 of peak heat release rates, , for vertical cabinets with more than one bundle of unqualified cable (NUREG/CR-6850, Appendix E). The probability density function (pdf) is denoted ; , , where and (2.6 and 67.8) are parameters of this particular gamma distribution.
13 NUREG/CR-6850 specifies gamma distributions for the various types of combustibles found within an NPP. Microsoft Excel®
provides a built-in function (GAMMA.DIST) that calculates the probability density function given the parameters and .
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Figure 4-4. Distribution of HRR for an electrical cabinet fire.
For convenience, a spreadsheet can be used to take each value of , from 1 kW to 600 kW, and compute a corresponding flame height, f (in meters), using Heskestad’s correlation:
f 0.235 / 1.02 (4-14)
The diameter of the fire, , is fixed at 0.48 m (1.6 ft), based on the equivalent diameter of the vent. Whereas Appendix E of NUREG/CR-6850 recommends dividing the range of into 15
“bins”, it is just as easy for this example to compute the flame height for 600 values of (each bin has a width of 1 kW). The pdf for the flame height, f , is related to the pdf for by the following expression:
f
; ,
f ; , /
0.094 (4-15)
This distribution is shown in Figure 4-5. Note that when the derivative in Equation (4-15) is not easily written in closed form, it is sufficient to calculate the bin width of the model output divided by that of the model input. In this example, the bin width of the model input parameter, , is 1 kW and the bin width of the model output parameter is the difference in flame heights for two successive values of .
HRR Distribution
Heat Release Rate (kW)
0 200 400 600
Probability Density Function
0.000 0.001 0.002 0.003 0.004 0.005
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Figure 4-5. Distribution of flame heights for the entire range of cabinet fires.
Once the pdf for the flame height is calculated, it can be used to determine the probability of the flames reaching a certain height. In this case, the cable tray is 1.5 m (4.9 ft) above the top of the cabinet. The probability that the flames from a randomly chosen fire will reach the cables is given by the area beneath the curve in Figure 4-5 for flame heights greater than 1.5 m (4.9 ft).
In this example, it is approximately 0.31.
4.4.2 Sensitivity Analysis
The algebraic calculations described in this report are perhaps the models more amenable to parameter uncertainty studies in practical applications. The more complex fire models discussed in this report can require dozens of physical and numerical input parameters for a given fire scenario. However, only a few of these parameters, when varied over their plausible range of values, will significantly impact the results. For example, the thermal conductivity of the compartment walls will not significantly affect a predicted cable surface temperature. Table 4-3 lists the input parameters whose impact on the given output quantity significantly outweighs all the other parameters. The HRR is almost always one of these.
In Volume 2 of NUREG-1824 (EPRI 1011999), Hamins quantifies the functional dependence of these key input parameters (see Table 4-3). These relationships are based either on the governing mathematical equations or on algebraic models. The basic mathematical form of the relationship is:
Output Quantity Constant × (Input Parameter)Power (4-16) The exact value of the Constant is not important; rather, it is the Power that matters. The larger its absolute value, the more important the Input Parameter. According to the McCaffrey,
Quintiere, and Harkleroad (MQH) correlation, for example, the hot gas layer (HGL) temperature rise in a compartment fire is proportional to the HRR raised to the two-thirds power:
/ (4-17)
What is important here is the amount that the HGL temperature, ∆ , changes due to a shift in the HRR, ∆ . It is the two-thirds power dependence, as found in Table 4-3, that matters. To
Flame Height Distribution
Flame Height (m)
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Probability Density Function
0.0 0.2 0.4 0.6 0.8 1.0
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see why, take the first derivative of with respect to and write the result in terms of differentials:
∆ 2
3
∆ (4-18)
This is a simple formula with which one can readily estimate the relative change in the model output quantity, ∆ / , due to the relative change in the model input parameter, ∆ / . The uncertainty in a measured quantity is often expressed in relative terms14. Suppose that the uncertainty in the HRR of the fire, ∆ / , is 0.15, or 15%. The expression above indicates that a 15% increase in the HRR should lead to a 2/3 x 15 = 10% increase in the prediction of the HGL temperature. The result is equally valid for a reduction; if the HRR is reduced by 15%, the HGL temperature is reduced by 10%.
Table 4-3. Sensitivity of model outputs from Volume 2 of NUREG-1824 (EPRI 1011999).
Output Quantity Important Input
Parameters Power Dependence
This relationship is based on experimental data, and has nothing to do with any particular model; however, an effective way to check a fire model is to take a simple compartment fire simulation, vary the HRR, and ensure that the change in the HGL temperature agrees with the correlation. Consider the two curves shown in Figure 4-6. For Benchmark Exercise #3 of the International Collaborative Fire Model Project (ICFMP), Test 3 was simulated with FDS, using HRR values of 1000 kW and 1150 kW. An examination of the peak values confirms that the relative change in the HGL temperature (10%) is two-thirds the relative change in the HRR
14 Note that a differential relationship is only approximate. This method of relating input parameters to output quantities is valid for relative differences that are less than approximately 30% in absolute value.
UNCERTAINTY AND SENSITIVITY (15%), consistent with the empirical result of the MQH correlation. Even though FDS is a much more complicated model than the simple expression shown above, it still exhibits the same functional dependence on the HRR.
Figure 4-6. FDS predictions of HGL Temperature as a function of time due to a 1,000 kW fire (solid line) and a 1,150 kW fire (dashed).
This section illustrates the usefulness of sensitivity analysis. As an example, consider that NFPA 805 uses the term Maximum Expected Fire Scenario (MEFS) to describe a severe fire that could be “reasonably anticipated” to occur within a compartment and the term Limiting Fire Scenario (LFS) to describe a severe fire that exceeds one or more performance criteria. The analyst is often asked to determine the model inputs for both of these scenarios. For MEFS, input parameters can be chosen from distributions for a particular percentile value. Gallucci (2011) discusses the issues important in determining the appropriate choice of HRR distribution.
The development of the LFS is essentially a sensitivity analysis performed to identify which combinations of input parameters or variables are critical to the analysis. The particular variables to be evaluated depend entirely on the problem being analyzed. At a minimum, the following parameters should be varied until failure conditions result: HRR, the fire growth rate or the flame spread rate, the flame radiative fraction or the radiative power, and the location of the fuel package relative to the target (if variable).
Suppose, for example, that as part of an analysis the problem is to determine the minimum HRR needed to cause damage in a particular compartment whose HGL temperature is not to exceed 500 °C (930 °F). The geometrical complexity of the compartment rules out the use of the algebraic and zone models, and that FDS has been selected for the simulation.
Step 1: Determine a reasonable, but conservative, estimate of what might be the maximum fire that could occur in the compartment. Using data from NUREG/CR-6850, for this example, suppose that a 98th percentile HRR for the electrical cabinet fire, 702 kW, has been chosen for this representative estimate. Choose a model and calculate the peak HGL temperature.
Step 2: The model chosen is FDS and it predicts 450 °C (840 °F) for the selected fire scenario.
Adjust the prediction to account for the model bias, (see Table 4-1):
Hot Gas Layer Temperature ICFMP BE #3, Test 3
Time (min)
0 5 10 15 20 25 30
Temperature (°C)
0 100 200 300 400
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adj 20 450 20
1.03 437 (4-19)
Step 3: Calculate the change in HRR required to increase the HGL temperature to 500 °C (930 °F):
This calculation suggests that adding an additional 159 kW to the original 702 kW will produce an HGL temperature in the vicinity of 500 °C (930 °F). This result can be double-checked by re-running the model with the modified input parameters.
Table 4-3 lists several other parameters besides the HRR that can affect the HGL temperature.
Following the example just discussed, similar calculations can be performed in which these other parameters are varied to determine how else the minimum damage threshold might be reached. For example, suppose that the surface area, , of the compartment is 400 m2 (4300 ft2). How much would the surface area have to increase or decrease to raise the HGL
temperature to 500 °C (930 °F)? If the thermal conductivity of the walls, , is 0.1 W/m/K, how much would it have to change? If the ventilation rate is 1 m3/s (2100 cfm), how much would it have to change? If the door height, , is 2 m (6.6 ft), how much would it have to change?
Following the example for the HRR, the required changes in these parameters can be calculated as follows:
For this example, to increase the HGL temperature by 63 °C (145 °F), one could increase the HRR by 159 kW, decrease the surface area of the compartment by 181 m2 (1948 ft2), decrease the thermal conductivity of the walls by 0.045 W/m/K, decrease the ventilation rate by 0.45 m3/s (950 cfm), or decrease the door height by 1.8 m (5.9 ft). Of course, some of these options are not physically possible. Room dimensions and thermal properties are not subject to significant change, but the HRR and ventilation rates can vary significantly. Also note that if the relative change in the parameter values exceeds 30 %, it is recommended that further calculations be performed to confirm the estimated quantity changes.
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